From 6e104e004f9514be0e016c8807dd919380dcbc3e Mon Sep 17 00:00:00 2001 From: "Daniel J. McDonald" Date: Mon, 18 Sep 2023 17:37:19 -0700 Subject: [PATCH] revise 09 lasso --- .../execute-results/html.json | 4 +- .../figure-revealjs/unnamed-chunk-1-1.png | Bin 220704 -> 225226 bytes .../figure-revealjs/unnamed-chunk-2-1.png | Bin 61973 -> 61818 bytes .../figure-revealjs/unnamed-chunk-3-1.svg | 16 +- .../09-l1-penalties/execute-results/html.json | 20 + .../figure-revealjs/convexity-1.svg | 1185 ++++++++++++++ .../figure-revealjs/plotting-functions-1.svg | 796 ++++++++++ .../figure-revealjs/ridge-v-lasso-1.svg | 275 ++++ .../figure-revealjs/ridge-v-lasso-again-1.svg | 296 ++++ .../figure-revealjs/unnamed-chunk-1-1.svg | 321 ++++ .../figure-revealjs/unnamed-chunk-3-1.svg | 580 +++++++ .../figure-revealjs/unnamed-chunk-5-1.svg | 567 +++++++ .../figure-revealjs/unnamed-chunk-7-1.svg | 1373 +++++++++++++++++ schedule/slides/08-ridge-regression.qmd | 203 +-- schedule/slides/09-l1-penalties.Rmd | 370 ----- schedule/slides/09-l1-penalties.html | 482 ------ schedule/slides/09-l1-penalties.qmd | 414 +++++ schedule/slides/_titleslide.qmd | 6 + 18 files changed, 5958 insertions(+), 950 deletions(-) create mode 100644 _freeze/schedule/slides/09-l1-penalties/execute-results/html.json create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/convexity-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/plotting-functions-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/ridge-v-lasso-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/ridge-v-lasso-again-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/unnamed-chunk-1-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/unnamed-chunk-3-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/unnamed-chunk-5-1.svg create mode 100644 _freeze/schedule/slides/09-l1-penalties/figure-revealjs/unnamed-chunk-7-1.svg delete mode 100644 schedule/slides/09-l1-penalties.Rmd delete mode 100644 schedule/slides/09-l1-penalties.html create mode 100644 schedule/slides/09-l1-penalties.qmd diff --git a/_freeze/schedule/slides/08-ridge-regression/execute-results/html.json b/_freeze/schedule/slides/08-ridge-regression/execute-results/html.json index 1512644..d71d86e 100644 --- a/_freeze/schedule/slides/08-ridge-regression/execute-results/html.json +++ b/_freeze/schedule/slides/08-ridge-regression/execute-results/html.json @@ -1,7 +1,7 @@ { - "hash": "7229912992807aad57117f3aa1d526f4", + "hash": "aee44d5a3c8bb5d5097f8c356a92a2c9", "result": { - "markdown": "---\nlecture: \"08 Ridge regression\"\nformat: revealjs\nmetadata-files: \n - _metadata.yml\n---\n---\n---\n\n## {{< meta lecture >}} {.large background-image=\"gfx/smooths.svg\" background-opacity=\"0.3\"}\n\n[Stat 406]{.secondary}\n\n[{{< meta author >}}]{.secondary}\n\nLast modified -- 08 September 2023\n\n\n\n$$\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\minimize}{minimize}\n\\DeclareMathOperator*{\\maximize}{maximize}\n\\DeclareMathOperator*{\\find}{find}\n\\DeclareMathOperator{\\st}{subject\\,\\,to}\n\\newcommand{\\E}{E}\n\\newcommand{\\Expect}[1]{\\E\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[2]{\\mathrm{Cov}\\left[#1,\\ #2\\right]}\n\\newcommand{\\given}{\\ \\vert\\ }\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\y}{\\mathbf{y}}\n\\newcommand{\\P}{\\mathcal{P}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\norm}[1]{\\left\\lVert #1 \\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert #1 \\rVert}\n$$\n\n\n\n\n\n## Recap\n\nSo far, we have emphasized __model selection__ as\n\n[Decide which predictors we would like to use in our linear model]{.hand}\n\nOr similarly:\n\n[Decide which of a few linear models to use]{.hand}\n\nTo do this, we used a risk estimate, and chose the \"model\" with the lowest estimate\n\n. . .\n\nMoving forward, we need to generalize this to\n\n[Decide which of possibly infinite prediction functions]{.hand} $f\\in\\mathcal{F}$ [to use]{.hand}\n\nThankfully, this isn't really any different. We still use those same risk estimates.\n\n\n[Remember:]{.secondary} We were choosing models that balance bias and variance (and hence have low prediction risk).\n\n$$\n\\newcommand{\\brt}{\\widehat{\\beta}^R_{s}}\n\\newcommand{\\brl}{\\widehat{\\beta}^R_{\\lambda}}\n\\newcommand{\\bls}{\\widehat{\\beta}_{ols}}\n\\newcommand{\\blt}{\\widehat{\\beta}^L_{s}}\n\\newcommand{\\bll}{\\widehat{\\beta}^L_{\\lambda}}\n$$\n\n\n\n## Regularization\n\n\n* Another way to control bias and variance is through [regularization]{.secondary} or\n[shrinkage]{.secondary}. \n\n\n* Rather than selecting a few predictors that seem reasonable, maybe trying a few combinations, use them all.\n\n* I mean [ALL]{.tertiary}.\n\n* But, make your estimates of $\\beta$ \"smaller\"\n\n\n\n## Brief aside on optimization {background-color=\"#97D4E9\"}\n\n* An optimization problem has 2 components:\n\n 1. The \"Objective function\": e.g. $\\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$.\n 2. The \"constraint\": e.g. \"fewer than 5 non-zero entries in $\\beta$\".\n \n* A constrained minimization problem is written\n\n\n$$\\min_\\beta f(\\beta)\\;\\; \\mbox{ subject to }\\;\\; C(\\beta)$$\n\n* $f(\\beta)$ is the objective function\n* $C(\\beta)$ is the constraint\n\n\n## Ridge regression (constrained version)\n\nOne way to do this for regression is to solve (say):\n$$\n\\minimize_\\beta \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2\n\\quad \\st \\sum_j \\beta^2_j < s\n$$\nfor some $s>0$.\n\n* This is called \"ridge regression\".\n* Call the [minimizer]{.secondary} of this problem $\\brt$\n\n. . .\n\nCompare this to ordinary least squares:\n\n$$\n\\minimize_\\beta \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 \n\\quad \\st \\beta \\in \\R^p\n$$\n\n## Visualizing ridge regression (2 coefficients)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nb <- c(1, 1)\nn <- 1000\nlams <- c(1, 5, 10)\nols_loss <- function(b1, b2) colMeans((y - X %*% rbind(b1, b2))^2) / 2\npen <- function(b1, b2, lambda = 1) sqrt(n) * lambda * (b1^2 + b2^2) / 2\ngr <- expand_grid(\n b1 = seq(b[1] - 0.5, b[1] + 0.5, length.out = 100),\n b2 = seq(b[2] - 0.5, b[2] + 0.5, length.out = 100)\n)\n\nX <- mvtnorm::rmvnorm(n, c(0, 0), sigma = matrix(c(1, .3, .3, .5), nrow = 2))\ny <- drop(X %*% b + rnorm(n))\n\nbols <- coef(lm(y ~ X - 1))\nbridge <- coef(MASS::lm.ridge(y ~ X - 1, lambda = lams * sqrt(n)))\n\npenalties <- lams |>\n set_names(~ paste(\"lam =\", .)) |>\n map(~ pen(gr$b1, gr$b2, .x)) |>\n as_tibble()\ngr <- gr |>\n mutate(loss = ols_loss(b1, b2)) |>\n bind_cols(penalties)\n\ng1 <- ggplot(gr, aes(b1, b2)) +\n geom_raster(aes(fill = loss)) +\n scale_fill_viridis_c(direction = -1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 20, barheight = 0.5))\n\ng2 <- gr |>\n pivot_longer(starts_with(\"lam\")) |>\n mutate(name = factor(name, levels = paste(\"lam =\", lams))) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = value)) +\n scale_fill_viridis_c(direction = -1, name = \"penalty\") +\n facet_wrap(~name, ncol = 1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5))\n\ng3 <- gr |> \n mutate(across(starts_with(\"lam\"), ~ loss + .x)) |>\n pivot_longer(starts_with(\"lam\")) |>\n mutate(name = factor(name, levels = paste(\"lam =\", lams))) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = value)) +\n scale_fill_viridis_c(direction = -1, name = \"loss + pen\") +\n facet_wrap(~name, ncol = 1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5))\n\ncowplot::plot_grid(g1, g2, g3, rel_widths = c(2, 1, 1), nrow = 1)\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-1-1.png){fig-align='center'}\n:::\n:::\n\n\n## The effect on the estimates\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\ngr |> \n mutate(z = ols_loss(b1, b2) + max(lams) * pen(b1, b2)) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = z)) +\n scale_fill_viridis_c(direction = -1) +\n geom_point(data = tibble(\n b1 = c(bols[1], bridge[,1]),\n b2 = c(bols[2], bridge[,2]),\n estimate = factor(c(\"ols\", paste0(\"ridge = \", lams)), \n levels = c(\"ols\", paste0(\"ridge = \", lams)))\n ),\n aes(shape = estimate), size = 3) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2]), colour = orange, size = 4)\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-2-1.png){fig-align='center'}\n:::\n:::\n\n\n\n## Geometry of ridge regression (contours)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nlibrary(mvtnorm)\nnorm_ball <- function(q = 1, len = 1000) {\n tg <- seq(0, 2 * pi, length = len)\n out <- tibble(x = cos(tg), b = (1 - abs(x)^q)^(1 / q), bm = -b) |>\n pivot_longer(-x, values_to = \"y\")\n out$lab <- paste0('\"||\" * beta * \"||\"', \"[\", signif(q, 2), \"]\")\n return(out)\n}\n\nellipse_data <- function(\n n = 75, xlim = c(-2, 3), ylim = c(-2, 3),\n mean = c(1, 1), Sigma = matrix(c(1, 0, 0, .5), 2)) {\n expand_grid(\n x = seq(xlim[1], xlim[2], length.out = n),\n y = seq(ylim[1], ylim[2], length.out = n)) |>\n rowwise() |>\n mutate(z = dmvnorm(c(x, y), mean, Sigma))\n}\n\nlballmax <- function(ed, q = 1, tol = 1e-6, niter = 20) {\n ed <- filter(ed, x > 0, y > 0)\n feasible <- (ed$x^q + ed$y^q)^(1 / q) <= 1\n best <- ed[feasible, ]\n best[which.max(best$z), ]\n}\n\n\nnb <- norm_ball(2)\ned <- ellipse_data()\nbols <- data.frame(x = 1, y = 1)\nbhat <- lballmax(ed, 2)\nggplot(nb, aes(x, y)) +\n xlim(-2, 2) +\n ylim(-2, 2) +\n geom_path(color = red) +\n geom_contour(mapping = aes(z = z), color = blue, data = ed, bins = 7) +\n geom_vline(xintercept = 0) +\n geom_hline(yintercept = 0) +\n geom_point(data = bols) +\n coord_equal() +\n geom_label(\n data = bols,\n mapping = aes(label = bquote(\"hat(beta)[ols]\")),\n parse = TRUE, \n nudge_x = .3, nudge_y = .3\n ) +\n geom_point(data = bhat) +\n xlab(bquote(beta[1])) +\n ylab(bquote(beta[2])) +\n theme_bw(base_size = 24) +\n geom_label(\n data = bhat,\n mapping = aes(label = bquote(\"hat(beta)[s]^R\")),\n parse = TRUE,\n nudge_x = -.4, nudge_y = -.4\n )\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/plotting-functions-1.svg){fig-align='center'}\n:::\n:::\n\n\n## Brief aside on norms\n\nRecall, for a vector $z \\in \\R^p$\n\n\n$$\\snorm{z}_2 = \\sqrt{z_1^2 + z_2^2 + \\cdots + z^2_p} = \\sqrt{\\sum_{j=1}^p z_j^2}$$\n\n\nSo, \n\n$$\\snorm{z}^2_2 = z_1^2 + z_2^2 + \\cdots + z^2_p = \\sum_{j=1}^p z_j^2.$$\n\n\n## Other norms we should remember:\n\n$\\ell_q$-norm\n: $\\left(\\sum_{j=1}^p |z_j|^q\\right)^{1/q}$\n\n$\\ell_1$-norm (special case)\n: $\\sum_{j=1}^p |z_j|$\n\n$\\ell_0$-norm\n: $\\sum_{j=1}^p I(z_j \\neq 0 ) = \\lvert \\{j : z_j \\neq 0 \\}\\rvert$\n\n\n\n## Ridge regression\n\nAn equivalent way to write\n\n$$\\brt = \\argmin_{ || \\beta ||_2^2 \\leq s} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$$\n\n\nis in the [Lagrangian]{.secondary} form\n\n\n$$\\brl = \\argmin_{ \\beta} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 + \\lambda || \\beta ||_2^2.$$\n\n\n\n\nFor every $\\lambda$ there is a unique $s$ (and vice versa) that makes \n\n$$\\brt = \\brl$$\n\n## Ridge regression\n\n$\\brt = \\argmin_{ || \\beta ||_2^2 \\leq s} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$\n\n$\\brl = \\argmin_{ \\beta} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 + \\lambda || \\beta ||_2^2$\n\nObserve:\n\n* $\\lambda = 0$ (or $s = \\infty$) makes $\\brl = \\bls$\n* Any $\\lambda > 0$ (or $s <\\infty$) penalizes larger values of $\\beta$, effectively shrinking them.\n\n\n$\\lambda$ and $s$ are known as [tuning parameters]{.secondary}\n\n\n## Example data\n\n`prostate` data from [ESL]\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\ndata(prostate, package = \"ElemStatLearn\")\nprostate |> as_tibble()\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n# A tibble: 97 × 10\n lcavol lweight age lbph svi lcp gleason pgg45 lpsa train\n \n 1 -0.580 2.77 50 -1.39 0 -1.39 6 0 -0.431 TRUE \n 2 -0.994 3.32 58 -1.39 0 -1.39 6 0 -0.163 TRUE \n 3 -0.511 2.69 74 -1.39 0 -1.39 7 20 -0.163 TRUE \n 4 -1.20 3.28 58 -1.39 0 -1.39 6 0 -0.163 TRUE \n 5 0.751 3.43 62 -1.39 0 -1.39 6 0 0.372 TRUE \n 6 -1.05 3.23 50 -1.39 0 -1.39 6 0 0.765 TRUE \n 7 0.737 3.47 64 0.615 0 -1.39 6 0 0.765 FALSE\n 8 0.693 3.54 58 1.54 0 -1.39 6 0 0.854 TRUE \n 9 -0.777 3.54 47 -1.39 0 -1.39 6 0 1.05 FALSE\n10 0.223 3.24 63 -1.39 0 -1.39 6 0 1.05 FALSE\n# ℹ 87 more rows\n```\n:::\n:::\n\n\n::: notes\n\nUse `lpsa` as response.\n\n:::\n\n\n## Ridge regression path\n\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\nY <- prostate$lpsa\nX <- model.matrix(~ ., data = prostate |> dplyr::select(-train, -lpsa))\nlibrary(glmnet)\nridge <- glmnet(x = X, y = Y, alpha = 0, lambda.min.ratio = .00001)\n```\n:::\n\n\n::: flex\n::: w-60\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\nplot(ridge, xvar = \"lambda\")\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-3-1.svg){fig-align='center'}\n:::\n:::\n\n\n:::\n::: w-35\n\nModel selection here: \n\n* means [choose]{.secondary} some $\\lambda$ \n\n* A value of $\\lambda$ is a vertical line.\n\n* This graphic is a \"path\" or \"coefficient trace\"\n\n* Coefficients for varying $\\lambda$\n:::\n:::\n\n\n## Solving the minimization\n\n* One nice thing about ridge regression is that it has a closed-form solution (like OLS)\n\n\n$$\\brl = (\\X^\\top\\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y$$\n\n* This is easy to calculate in `R` for any $\\lambda$.\n\n* However, computations and interpretation are simplified if we examine the Singular Value Decomposition of $\\X = \\mathbf{UDV}^\\top$.\n\n* Recall: any matrix has an SVD.\n\n* Here $\\mathbf{D}$ is diagonal and $\\mathbf{U}$ and $\\mathbf{V}$ are orthonormal: $\\mathbf{U}^\\top\\mathbf{U} = \\mathbf{I}$.\n\n## Solving the minization\n\n$$\\brl = (\\X^\\top\\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y$$\n\n* Note that $\\mathbf{X}^\\top\\mathbf{X} = \\mathbf{VDU}^\\top\\mathbf{UDV}^\\top = \\mathbf{V}\\mathbf{D}^2\\mathbf{V}^\\top$.\n\n\n* Then,\n\n\n$$\\brl = (\\X^\\top \\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y = (\\mathbf{VD}^2\\mathbf{V}^\\top + \\lambda \\mathbf{I})^{-1}\\mathbf{VDU}^\\top \\y\n= \\mathbf{V}(\\mathbf{D}^2+\\lambda \\mathbf{I})^{-1} \\mathbf{DU}^\\top \\y.$$\n\n* For computations, now we only need to invert $\\mathbf{D}$.\n\n\n## Comparing with OLS\n\n\n* $\\mathbf{D}$ is a diagonal matrix\n\n$$\\bls = (\\X^\\top\\X)^{-1}\\X^\\top \\y = (\\mathbf{VD}^2\\mathbf{V}^\\top)^{-1}\\mathbf{VDU}^\\top \\y = \\mathbf{V}\\color{red}{\\mathbf{D}^{-2}\\mathbf{D}}\\mathbf{U}^\\top \\y = \\mathbf{V}\\color{red}{\\mathbf{D}^{-1}}\\mathbf{U}^\\top \\y$$\n\n$$\\brl = (\\X^\\top \\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y = \\mathbf{V}\\color{red}{(\\mathbf{D}^2+\\lambda \\mathbf{I})^{-1}} \\mathbf{DU}^\\top \\y.$$\n\n\n* Notice that $\\bls$ depends on $d_j/d_j^2$ while $\\brl$ depends on $d_j/(d_j^2 + \\lambda)$.\n\n* Ridge regression makes the coefficients smaller relative to OLS.\n\n* But if $\\X$ has small singular values, ridge regression compensates with $\\lambda$ in the denominator.\n\n\n\n## Ridge regression and multicollinearity\n\n[Multicollinearity:]{.secondary} a linear combination of predictor variables is nearly equal to another predictor variable. \n\nSome comments:\n\n* A better phrase: $\\X$ is ill-conditioned\n\n* AKA \"(numerically) rank-deficient\".\n\n* $\\X = \\mathbf{U D V}^\\top$ ill-conditioned $\\Longleftrightarrow$ some elements of $\\mathbf{D} \\approx 0$\n\n* $\\bls= \\mathbf{V D}^{-1} \\mathbf{U}^\\top \\y$, so small entries of $\\mathbf{D}$ $\\Longleftrightarrow$ huge elements of $\\mathbf{D}^{-1}$\n\n* Means huge variance: $\\Var{\\bls} = \\sigma^2(\\X^\\top \\X)^{-1} = \\sigma^2 \\mathbf{V D}^{-2} \\mathbf{V}^\\top$\n\n\n## Ridge regression and ill-posed $\\X$\n\n\nRidge Regression fixes this problem by preventing the division by a near-zero number\n\nConclusion\n: $(\\X^{\\top}\\X)^{-1}$ can be really unstable, while $(\\X^{\\top}\\X + \\lambda \\mathbf{I})^{-1}$ is not.\n\nAside\n: Engineering approach to solving linear systems is to always do this with small $\\lambda$. The thinking is about the numerics rather than the statistics.\n\n\n## Can we get the best of both worlds?\n\nTo recap:\n\n* Deciding which predictors to include, adding quadratic terms, or interactions is [model selection]{.secondary} (more precisely variable selection within a linear model).\n\n* Ridge regression provides regularization, which trades off bias and variance and also stabilizes multicollinearity. \n\n* If the LM is **true**, \n 1. OLS is unbiased, but Variance depends on $\\mathbf{D}^{-2}$. Can be big.\n 2. Ridge is biased (can you find the bias?). But Variance is smaller than OLS.\n\n* Ridge regression does not perform variable selection.\n\n* But [picking]{.hand} $\\lambda=3.7$ and thereby .hand[deciding] to predict with $\\widehat{\\beta}^R_{3.7}$ is [model selection]{.secondary}.\n\n## Can we get the best of both worlds?\n\nRidge regression \n: $\\minimize \\frac{1}{n}||\\y-\\X\\beta||_2^2 \\ \\st\\ ||\\beta||_2^2 \\leq s$ \n\nBest (in-sample) linear regression model of size $s$\n: $\\minimize \\frac{1}{n}||\\y-\\X\\beta||_2^2 \\ \\st\\ ||\\beta||_0 \\leq s$\n\n\n$||\\beta||_0$ is the number of nonzero elements in $\\beta$\n\nFinding the best in-sample linear model (of size $s$, among these predictors) is a nonconvex optimization problem (In fact, it is NP-hard)\n\nRidge regression is convex (easy to solve), but doesn't do __variable__ selection\n\nCan we somehow \"interpolate\" to get both?\n\n\nNote: selecting $\\lambda$ is still __model__ selection, but we've included __all__ the variables.\n\n\n# Next time...\n\nThe lasso, interpolating variable selection and model selection\n", + "markdown": "---\nlecture: \"08 Ridge regression\"\nformat: revealjs\nmetadata-files: \n - _metadata.yml\n---\n---\n---\n\n## {{< meta lecture >}} {.large background-image=\"gfx/smooths.svg\" background-opacity=\"0.3\"}\n\n[Stat 406]{.secondary}\n\n[{{< meta author >}}]{.secondary}\n\nLast modified -- 18 September 2023\n\n\n\n$$\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\minimize}{minimize}\n\\DeclareMathOperator*{\\maximize}{maximize}\n\\DeclareMathOperator*{\\find}{find}\n\\DeclareMathOperator{\\st}{subject\\,\\,to}\n\\newcommand{\\E}{E}\n\\newcommand{\\Expect}[1]{\\E\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[2]{\\mathrm{Cov}\\left[#1,\\ #2\\right]}\n\\newcommand{\\given}{\\ \\vert\\ }\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\y}{\\mathbf{y}}\n\\newcommand{\\P}{\\mathcal{P}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\norm}[1]{\\left\\lVert #1 \\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert #1 \\rVert}\n$$\n\n\n\n\n\n## Recap\n\nSo far, we have emphasized __model selection__ as\n\n[Decide which predictors we would like to use in our linear model]{.hand}\n\nOr similarly:\n\n[Decide which of a few linear models to use]{.hand}\n\nTo do this, we used a risk estimate, and chose the \"model\" with the lowest estimate\n\n. . .\n\nMoving forward, we need to generalize this to\n\n[Decide which of possibly infinite prediction functions]{.hand} $f\\in\\mathcal{F}$ [to use]{.hand}\n\nThankfully, this isn't really any different. We still use those same risk estimates.\n\n\n[Remember:]{.secondary} We were choosing models that balance bias and variance (and hence have low prediction risk).\n\n$$\n\\newcommand{\\brt}{\\widehat{\\beta}^R_{s}}\n\\newcommand{\\brl}{\\widehat{\\beta}^R_{\\lambda}}\n\\newcommand{\\bls}{\\widehat{\\beta}_{ols}}\n\\newcommand{\\blt}{\\widehat{\\beta}^L_{s}}\n\\newcommand{\\bll}{\\widehat{\\beta}^L_{\\lambda}}\n$$\n\n\n\n## Regularization\n\n\n* Another way to control bias and variance is through [regularization]{.secondary} or\n[shrinkage]{.secondary}. \n\n\n* Rather than selecting a few predictors that seem reasonable, maybe trying a few combinations, use them all.\n\n* I mean [ALL]{.tertiary}.\n\n* But, make your estimates of $\\beta$ \"smaller\"\n\n\n\n## Brief aside on optimization {background-color=\"#97D4E9\"}\n\n* An optimization problem has 2 components:\n\n 1. The \"Objective function\": e.g. $\\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$.\n 2. The \"constraint\": e.g. \"fewer than 5 non-zero entries in $\\beta$\".\n \n* A constrained minimization problem is written\n\n\n$$\\min_\\beta f(\\beta)\\;\\; \\mbox{ subject to }\\;\\; C(\\beta)$$\n\n* $f(\\beta)$ is the objective function\n* $C(\\beta)$ is the constraint\n\n\n## Ridge regression (constrained version)\n\nOne way to do this for regression is to solve (say):\n$$\n\\minimize_\\beta \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2\n\\quad \\st \\sum_j \\beta^2_j < s\n$$\nfor some $s>0$.\n\n* This is called \"ridge regression\".\n* Call the [minimizer]{.secondary} of this problem $\\brt$\n\n. . .\n\nCompare this to ordinary least squares:\n\n$$\n\\minimize_\\beta \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 \n\\quad \\st \\beta \\in \\R^p\n$$\n\n\n\n## Geometry of ridge regression (contours)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nlibrary(mvtnorm)\nnorm_ball <- function(q = 1, len = 1000) {\n tg <- seq(0, 2 * pi, length = len)\n out <- tibble(x = cos(tg), b = (1 - abs(x)^q)^(1 / q), bm = -b) |>\n pivot_longer(-x, values_to = \"y\")\n out$lab <- paste0('\"||\" * beta * \"||\"', \"[\", signif(q, 2), \"]\")\n return(out)\n}\n\nellipse_data <- function(\n n = 75, xlim = c(-2, 3), ylim = c(-2, 3),\n mean = c(1, 1), Sigma = matrix(c(1, 0, 0, .5), 2)) {\n expand_grid(\n x = seq(xlim[1], xlim[2], length.out = n),\n y = seq(ylim[1], ylim[2], length.out = n)) |>\n rowwise() |>\n mutate(z = dmvnorm(c(x, y), mean, Sigma))\n}\n\nlballmax <- function(ed, q = 1, tol = 1e-6, niter = 20) {\n ed <- filter(ed, x > 0, y > 0)\n feasible <- (ed$x^q + ed$y^q)^(1 / q) <= 1\n best <- ed[feasible, ]\n best[which.max(best$z), ]\n}\n\n\nnb <- norm_ball(2)\ned <- ellipse_data()\nbols <- data.frame(x = 1, y = 1)\nbhat <- lballmax(ed, 2)\nggplot(nb, aes(x, y)) +\n xlim(-2, 2) +\n ylim(-2, 2) +\n geom_path(color = red) +\n geom_contour(mapping = aes(z = z), color = blue, data = ed, bins = 7) +\n geom_vline(xintercept = 0) +\n geom_hline(yintercept = 0) +\n geom_point(data = bols) +\n coord_equal() +\n geom_label(\n data = bols,\n mapping = aes(label = bquote(\"hat(beta)[ols]\")),\n parse = TRUE, \n nudge_x = .3, nudge_y = .3\n ) +\n geom_point(data = bhat) +\n xlab(bquote(beta[1])) +\n ylab(bquote(beta[2])) +\n theme_bw(base_size = 24) +\n geom_label(\n data = bhat,\n mapping = aes(label = bquote(\"hat(beta)[s]^R\")),\n parse = TRUE,\n nudge_x = -.4, nudge_y = -.4\n )\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/plotting-functions-1.svg){fig-align='center'}\n:::\n:::\n\n\n## Brief aside on norms\n\nRecall, for a vector $z \\in \\R^p$\n\n\n$$\\snorm{z}_2 = \\sqrt{z_1^2 + z_2^2 + \\cdots + z^2_p} = \\sqrt{\\sum_{j=1}^p z_j^2}$$\n\n\nSo, \n\n$$\\snorm{z}^2_2 = z_1^2 + z_2^2 + \\cdots + z^2_p = \\sum_{j=1}^p z_j^2.$$\n\n\n## Other norms we should remember:\n\n$\\ell_q$-norm\n: $\\left(\\sum_{j=1}^p |z_j|^q\\right)^{1/q}$\n\n$\\ell_1$-norm (special case)\n: $\\sum_{j=1}^p |z_j|$\n\n$\\ell_0$-norm\n: $\\sum_{j=1}^p I(z_j \\neq 0 ) = \\lvert \\{j : z_j \\neq 0 \\}\\rvert$\n\n$\\ell_\\infty$-norm\n: $\\max_{1\\leq j \\leq p} |z_j|$\n\n::: aside\nRecall what a norm is: \n:::\n\n\n## Ridge regression\n\nAn equivalent way to write\n\n$$\\brt = \\argmin_{ || \\beta ||_2^2 \\leq s} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$$\n\n\nis in the [Lagrangian]{.secondary} form\n\n\n$$\\brl = \\argmin_{ \\beta} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 + \\lambda || \\beta ||_2^2.$$\n\n\n\n\nFor every $\\lambda$ there is a unique $s$ (and vice versa) that makes \n\n$$\\brt = \\brl$$\n\n## Ridge regression\n\n$\\brt = \\argmin_{ || \\beta ||_2^2 \\leq s} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2$\n\n$\\brl = \\argmin_{ \\beta} \\frac{1}{n}\\sum_i (y_i-x^\\top_i \\beta)^2 + \\lambda || \\beta ||_2^2$\n\nObserve:\n\n* $\\lambda = 0$ (or $s = \\infty$) makes $\\brl = \\bls$\n* Any $\\lambda > 0$ (or $s <\\infty$) penalizes larger values of $\\beta$, effectively shrinking them.\n\n\n$\\lambda$ and $s$ are known as [tuning parameters]{.secondary}\n\n\n## Visualizing ridge regression (2 coefficients)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nb <- c(1, 1)\nn <- 1000\nlams <- c(1, 5, 10)\nols_loss <- function(b1, b2) colMeans((y - X %*% rbind(b1, b2))^2) / 2\npen <- function(b1, b2, lambda = 1) lambda * (b1^2 + b2^2) / 2\ngr <- expand_grid(\n b1 = seq(b[1] - 0.5, b[1] + 0.5, length.out = 100),\n b2 = seq(b[2] - 0.5, b[2] + 0.5, length.out = 100)\n)\n\nX <- mvtnorm::rmvnorm(n, c(0, 0), sigma = matrix(c(1, .3, .3, .5), nrow = 2))\ny <- drop(X %*% b + rnorm(n))\n\nbols <- coef(lm(y ~ X - 1))\nbridge <- coef(MASS::lm.ridge(y ~ X - 1, lambda = lams * sqrt(n)))\n\npenalties <- lams |>\n set_names(~ paste(\"lam =\", .)) |>\n map(~ pen(gr$b1, gr$b2, .x)) |>\n as_tibble()\ngr <- gr |>\n mutate(loss = ols_loss(b1, b2)) |>\n bind_cols(penalties)\n\ng1 <- ggplot(gr, aes(b1, b2)) +\n geom_raster(aes(fill = loss)) +\n scale_fill_viridis_c(direction = -1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 20, barheight = 0.5))\n\ng2 <- gr |>\n pivot_longer(starts_with(\"lam\")) |>\n mutate(name = factor(name, levels = paste(\"lam =\", lams))) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = value)) +\n scale_fill_viridis_c(direction = -1, name = \"penalty\") +\n facet_wrap(~name, ncol = 1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5))\n\ng3 <- gr |> \n mutate(across(starts_with(\"lam\"), ~ loss + .x)) |>\n pivot_longer(starts_with(\"lam\")) |>\n mutate(name = factor(name, levels = paste(\"lam =\", lams))) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = value)) +\n scale_fill_viridis_c(direction = -1, name = \"loss + pen\") +\n facet_wrap(~name, ncol = 1) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2])) +\n theme(legend.position = \"bottom\") +\n guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5))\n\ncowplot::plot_grid(g1, g2, g3, rel_widths = c(2, 1, 1), nrow = 1)\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-1-1.png){fig-align='center'}\n:::\n:::\n\n\n## The effect on the estimates\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\ngr |> \n mutate(z = ols_loss(b1, b2) + max(lams) * pen(b1, b2)) |>\n ggplot(aes(b1, b2)) +\n geom_raster(aes(fill = z)) +\n scale_fill_viridis_c(direction = -1) +\n geom_point(data = tibble(\n b1 = c(bols[1], bridge[,1]),\n b2 = c(bols[2], bridge[,2]),\n estimate = factor(c(\"ols\", paste0(\"ridge = \", lams)), \n levels = c(\"ols\", paste0(\"ridge = \", lams)))\n ),\n aes(shape = estimate), size = 3) +\n geom_point(data = data.frame(b1 = b[1], b2 = b[2]), colour = orange, size = 4)\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-2-1.png){fig-align='center'}\n:::\n:::\n\n\n\n## Example data\n\n`prostate` data from [ESL]\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\ndata(prostate, package = \"ElemStatLearn\")\nprostate |> as_tibble()\n```\n\n::: {.cell-output .cell-output-stdout}\n```\n# A tibble: 97 × 10\n lcavol lweight age lbph svi lcp gleason pgg45 lpsa train\n \n 1 -0.580 2.77 50 -1.39 0 -1.39 6 0 -0.431 TRUE \n 2 -0.994 3.32 58 -1.39 0 -1.39 6 0 -0.163 TRUE \n 3 -0.511 2.69 74 -1.39 0 -1.39 7 20 -0.163 TRUE \n 4 -1.20 3.28 58 -1.39 0 -1.39 6 0 -0.163 TRUE \n 5 0.751 3.43 62 -1.39 0 -1.39 6 0 0.372 TRUE \n 6 -1.05 3.23 50 -1.39 0 -1.39 6 0 0.765 TRUE \n 7 0.737 3.47 64 0.615 0 -1.39 6 0 0.765 FALSE\n 8 0.693 3.54 58 1.54 0 -1.39 6 0 0.854 TRUE \n 9 -0.777 3.54 47 -1.39 0 -1.39 6 0 1.05 FALSE\n10 0.223 3.24 63 -1.39 0 -1.39 6 0 1.05 FALSE\n# ℹ 87 more rows\n```\n:::\n:::\n\n\n::: notes\n\nUse `lpsa` as response.\n\n:::\n\n\n## Ridge regression path\n\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\nY <- prostate$lpsa\nX <- model.matrix(~ ., data = prostate |> dplyr::select(-train, -lpsa))\nlibrary(glmnet)\nridge <- glmnet(x = X, y = Y, alpha = 0, lambda.min.ratio = .00001)\n```\n:::\n\n\n
\n\n::: flex\n::: w-60\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\nplot(ridge, xvar = \"lambda\", lwd = 3)\n```\n\n::: {.cell-output-display}\n![](08-ridge-regression_files/figure-revealjs/unnamed-chunk-3-1.svg){fig-align='center'}\n:::\n:::\n\n\n:::\n::: w-35\n\nModel selection here: \n\n* means [choose]{.secondary} some $\\lambda$ \n\n* A value of $\\lambda$ is a vertical line.\n\n* This graphic is a \"path\" or \"coefficient trace\"\n\n* Coefficients for varying $\\lambda$\n:::\n:::\n\n\n## Solving the minimization\n\n* One nice thing about ridge regression is that it has a closed-form solution (like OLS)\n\n\n$$\\brl = (\\X^\\top\\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y$$\n\n* This is easy to calculate in `R` for any $\\lambda$.\n\n* However, computations and interpretation are simplified if we examine the \n[Singular Value Decomposition]{.secondary} of $\\X = \\mathbf{UDV}^\\top$.\n\n* Recall: any matrix has an SVD.\n\n* Here $\\mathbf{D}$ is diagonal and $\\mathbf{U}$ and $\\mathbf{V}$ are orthonormal: $\\mathbf{U}^\\top\\mathbf{U} = \\mathbf{I}$.\n\n## Solving the minization\n\n$$\\brl = (\\X^\\top\\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y$$\n\n* Note that $\\mathbf{X}^\\top\\mathbf{X} = \\mathbf{VDU}^\\top\\mathbf{UDV}^\\top = \\mathbf{V}\\mathbf{D}^2\\mathbf{V}^\\top$.\n\n\n* Then,\n\n\n$$\\brl = (\\X^\\top \\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y = (\\mathbf{VD}^2\\mathbf{V}^\\top + \\lambda \\mathbf{I})^{-1}\\mathbf{VDU}^\\top \\y\n= \\mathbf{V}(\\mathbf{D}^2+\\lambda \\mathbf{I})^{-1} \\mathbf{DU}^\\top \\y.$$\n\n* For computations, now we only need to invert $\\mathbf{D}$.\n\n\n## Comparing with OLS\n\n\n* $\\mathbf{D}$ is a diagonal matrix\n\n$$\\bls = (\\X^\\top\\X)^{-1}\\X^\\top \\y = (\\mathbf{VD}^2\\mathbf{V}^\\top)^{-1}\\mathbf{VDU}^\\top \\y = \\mathbf{V}\\color{red}{\\mathbf{D}^{-2}\\mathbf{D}}\\mathbf{U}^\\top \\y = \\mathbf{V}\\color{red}{\\mathbf{D}^{-1}}\\mathbf{U}^\\top \\y$$\n\n$$\\brl = (\\X^\\top \\X + \\lambda \\mathbf{I})^{-1}\\X^\\top \\y = \\mathbf{V}\\color{red}{(\\mathbf{D}^2+\\lambda \\mathbf{I})^{-1}} \\mathbf{DU}^\\top \\y.$$\n\n\n* Notice that $\\bls$ depends on $d_j/d_j^2$ while $\\brl$ depends on $d_j/(d_j^2 + \\lambda)$.\n\n* Ridge regression makes the coefficients smaller relative to OLS.\n\n* But if $\\X$ has small singular values, ridge regression compensates with $\\lambda$ in the denominator.\n\n\n\n## Ridge regression and multicollinearity\n\n[Multicollinearity:]{.secondary} a linear combination of predictor variables is nearly equal to another predictor variable. \n\nSome comments:\n\n* A better phrase: $\\X$ is ill-conditioned\n\n* AKA \"(numerically) rank-deficient\".\n\n* $\\X = \\mathbf{U D V}^\\top$ ill-conditioned $\\Longleftrightarrow$ some elements of $\\mathbf{D} \\approx 0$\n\n* $\\bls= \\mathbf{V D}^{-1} \\mathbf{U}^\\top \\y$, so small entries of $\\mathbf{D}$ $\\Longleftrightarrow$ huge elements of $\\mathbf{D}^{-1}$\n\n* Means huge variance: $\\Var{\\bls} = \\sigma^2(\\X^\\top \\X)^{-1} = \\sigma^2 \\mathbf{V D}^{-2} \\mathbf{V}^\\top$\n\n\n## Ridge regression and ill-posed $\\X$\n\n\nRidge Regression fixes this problem by preventing the division by a near-zero number\n\nConclusion\n: $(\\X^{\\top}\\X)^{-1}$ can be really unstable, while $(\\X^{\\top}\\X + \\lambda \\mathbf{I})^{-1}$ is not.\n\nAside\n: Engineering approach to solving linear systems is to always do this with small $\\lambda$. The thinking is about the numerics rather than the statistics.\n\n### Which $\\lambda$ to use?\n\nComputational\n: Use CV and pick the $\\lambda$ that makes this smallest.\n\nIntuition (bias)\n: As $\\lambda\\rightarrow\\infty$, bias ⬆\n\nIntuition (variance)\n: As $\\lambda\\rightarrow\\infty$, variance ⬇\n\nYou should think about why.\n\n\n\n## Can we get the best of both worlds?\n\nTo recap:\n\n* Deciding which predictors to include, adding quadratic terms, or interactions is [model selection]{.secondary} (more precisely variable selection within a linear model).\n\n* Ridge regression provides regularization, which trades off bias and variance and also stabilizes multicollinearity. \n\n* If the LM is **true**, \n 1. OLS is unbiased, but Variance depends on $\\mathbf{D}^{-2}$. Can be big.\n 2. Ridge is biased (can you find the bias?). But Variance is smaller than OLS.\n\n* Ridge regression does not perform variable selection.\n\n* But [picking]{.hand} $\\lambda=3.7$ and thereby [deciding]{.hand} to predict with $\\widehat{\\beta}^R_{3.7}$ is [model selection]{.secondary}.\n\n\n\n## Can we get the best of both worlds?\n\nRidge regression \n: $\\minimize \\frac{1}{n}||\\y-\\X\\beta||_2^2 \\ \\st\\ ||\\beta||_2^2 \\leq s$ \n\nBest (in-sample) linear regression model of size $s$\n: $\\minimize \\frac{1}{n}||\\y-\\X\\beta||_2^2 \\ \\st\\ ||\\beta||_0 \\leq s$\n\n\n$||\\beta||_0$ is the number of nonzero elements in $\\beta$\n\nFinding the best in-sample linear model (of size $s$, among these predictors) is a nonconvex optimization problem (In fact, it is NP-hard)\n\nRidge regression is convex (easy to solve), but doesn't do __variable__ selection\n\nCan we somehow \"interpolate\" to get both?\n\n\nNote: selecting $\\lambda$ is still __model__ selection, but we've included __all__ the variables.\n\n\n# Next time...\n\nThe lasso, interpolating variable selection and model selection\n", "supporting": [ "08-ridge-regression_files" ], diff --git a/_freeze/schedule/slides/08-ridge-regression/figure-revealjs/unnamed-chunk-1-1.png b/_freeze/schedule/slides/08-ridge-regression/figure-revealjs/unnamed-chunk-1-1.png index d3805662c27a300fc4e994b076a992d3b8740ae5..e9954c516c6754cf6c3e92c21c3b943d9a3aa94e 100644 GIT binary patch literal 225226 zcmbrmcUV)~wmuvXK|nwSK>-PZQlv=-DFKz<1f+NABmwC?5fMS@AWeiINbe;{Z>gp56aWwl57#lV zW7MIMPOd8Fde9R%l$8}jc_I9%0;1LbZer!ax*R&i9Sf?*YHEXA$abZhEp>cr8CTqL z^F;d_xUz?mY;Vj4dbu}BcD8OkKS%PLeIav)X??lAM8yP{`~8Tg#OH{WsYK2GQAk&~ zo}EbBUdZwHC0ZLo|1BO{`zND);+ZxdWx2OP=l5>vW=l8g<)$V)#hUR{(zHI5dT;vW zx?B5W_jAmthh9q3wcjz~bma755BYSFyHDHUdPw$fH_T9O@W|g&?P`;;odk$_1pVTR zSYYg?v2^q8#PAN8#&_Nuw;!8RKaXOjviCDG&&VdDU$OhTl4!J*J0jv`R;wBHm_NS_?7sSVgipDSz;F+-@Tu~9K+tB zZ53wK|3sNmxGkQ=)oQ=0f5@AFRSO{+S{u6JDcgEsYn3=x(c7Yo5WO+hJJt*Cj2m?{ zed6z^7g{5)i>MiKU3Gh4OYgwr);=+m;?j%zVYWP4Vzw-5c>RI|b_KpZsaq+$F7#8E z%)i>ns99lOO5I92lSc<)QJ#@*Agp^z2^Qpt>OlpYRN}R0n{R`gE<+^S-7LRYkegfH8sE^{P8saL9ji52!BL? 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+ + diff --git a/_freeze/schedule/slides/09-l1-penalties/execute-results/html.json b/_freeze/schedule/slides/09-l1-penalties/execute-results/html.json new file mode 100644 index 0000000..ea049b3 --- /dev/null +++ b/_freeze/schedule/slides/09-l1-penalties/execute-results/html.json @@ -0,0 +1,20 @@ +{ + "hash": "ce68719ed0f65c19464f8652dc6457be", + "result": { + "markdown": "---\nlecture: \"09 L1 penalties\"\nformat: revealjs\nmetadata-files: \n - _metadata.yml\n---\n---\n---\n\n## {{< meta lecture >}} {.large background-image=\"gfx/smooths.svg\" background-opacity=\"0.3\"}\n\n[Stat 406]{.secondary}\n\n[{{< meta author >}}]{.secondary}\n\nLast modified -- 18 September 2023\n\n\n\n$$\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\minimize}{minimize}\n\\DeclareMathOperator*{\\maximize}{maximize}\n\\DeclareMathOperator*{\\find}{find}\n\\DeclareMathOperator{\\st}{subject\\,\\,to}\n\\newcommand{\\E}{E}\n\\newcommand{\\Expect}[1]{\\E\\left[ #1 \\right]}\n\\newcommand{\\Var}[1]{\\mathrm{Var}\\left[ #1 \\right]}\n\\newcommand{\\Cov}[2]{\\mathrm{Cov}\\left[#1,\\ #2\\right]}\n\\newcommand{\\given}{\\ \\vert\\ }\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\y}{\\mathbf{y}}\n\\newcommand{\\P}{\\mathcal{P}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\norm}[1]{\\left\\lVert #1 \\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert #1 \\rVert}\n\\newcommand{\\tr}[1]{\\mbox{tr}(#1)}\n\\newcommand{\\brt}{\\widehat{\\beta}^R_{s}}\n\\newcommand{\\brl}{\\widehat{\\beta}^R_{\\lambda}}\n\\newcommand{\\bls}{\\widehat{\\beta}_{ols}}\n\\newcommand{\\blt}{\\widehat{\\beta}^L_{s}}\n\\newcommand{\\bll}{\\widehat{\\beta}^L_{\\lambda}}\n$$\n\n\n\n\n\n## Last time\n\n\nRidge regression\n: $\\min \\frac{1}{n}\\snorm{\\y-\\X\\beta}_2^2 \\st \\snorm{\\beta}_2^2 \\leq s$ \n\nBest (in sample) linear regression model of size $s$\n: $\\min \\frac 1n \\snorm{\\y-\\X\\beta}_2^2 \\st \\snorm{\\beta}_0 \\leq s$\n\n$\\snorm{\\beta}_0$ is the number of nonzero elements in $\\beta$\n\nFinding the \"best\" linear model (of size $s$, among these predictors, in sample) is a nonconvex optimization problem (In fact, it is NP-hard)\n\nRidge regression is convex (easy to solve), but doesn't do variable selection\n\nCan we somehow \"interpolate\" to get both?\n\n\n\n## Geometry of convexity\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nlibrary(mvtnorm)\nnormBall <- function(q = 1, len = 1000) {\n tg <- seq(0, 2 * pi, length = len)\n out <- data.frame(x = cos(tg)) %>%\n mutate(b = (1 - abs(x)^q)^(1 / q), bm = -b) %>%\n gather(key = \"lab\", value = \"y\", -x)\n out$lab <- paste0('\"||\" * beta * \"||\"', \"[\", signif(q, 2), \"]\")\n return(out)\n}\n\nellipseData <- function(n = 100, xlim = c(-2, 3), ylim = c(-2, 3),\n mean = c(1, 1), Sigma = matrix(c(1, 0, 0, .5), 2)) {\n df <- expand.grid(\n x = seq(xlim[1], xlim[2], length.out = n),\n y = seq(ylim[1], ylim[2], length.out = n)\n )\n df$z <- dmvnorm(df, mean, Sigma)\n df\n}\n\nlballmax <- function(ed, q = 1, tol = 1e-6) {\n ed <- filter(ed, x > 0, y > 0)\n for (i in 1:20) {\n ff <- abs((ed$x^q + ed$y^q)^(1 / q) - 1) < tol\n if (sum(ff) > 0) break\n tol <- 2 * tol\n }\n best <- ed[ff, ]\n best[which.max(best$z), ]\n}\n\nnbs <- list()\nnbs[[1]] <- normBall(0, 1)\nqs <- c(.5, .75, 1, 1.5, 2)\nfor (ii in 2:6) nbs[[ii]] <- normBall(qs[ii - 1])\nnbs <- bind_rows(nbs)\nnbs$lab <- factor(nbs$lab, levels = unique(nbs$lab))\nseg <- data.frame(\n lab = levels(nbs$lab)[1],\n x0 = c(-1, 0), x1 = c(1, 0), y0 = c(0, -1), y1 = c(0, 1)\n)\nlevels(seg$lab) <- levels(nbs$lab)\nggplot(nbs, aes(x, y)) +\n geom_path(size = 1.2) +\n facet_wrap(~lab, labeller = label_parsed) +\n geom_segment(data = seg, aes(x = x0, xend = x1, y = y0, yend = y1), size = 1.2) +\n theme_bw(base_family = \"\", base_size = 24) +\n coord_equal() +\n scale_x_continuous(breaks = c(-1, 0, 1)) +\n scale_y_continuous(breaks = c(-1, 0, 1)) +\n geom_vline(xintercept = 0, size = .5) +\n geom_hline(yintercept = 0, size = .5) +\n xlab(bquote(beta[1])) +\n ylab(bquote(beta[2]))\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/plotting-functions-1.svg){fig-align='center'}\n:::\n:::\n\n\n\n## The best of both worlds\n\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nnb <- normBall(1)\ned <- ellipseData()\nbols <- data.frame(x = 1, y = 1)\nbhat <- lballmax(ed, 1)\nggplot(nb, aes(x, y)) +\n geom_path(color = red) +\n geom_contour(mapping = aes(z = z), color = blue, data = ed, bins = 7) +\n geom_vline(xintercept = 0) +\n geom_hline(yintercept = 0) +\n geom_point(data = bols) +\n coord_equal(xlim = c(-2, 2), ylim = c(-2, 2)) +\n theme_bw(base_family = \"\", base_size = 24) +\n geom_label(\n data = bols, mapping = aes(label = bquote(\"hat(beta)[ols]\")), parse = TRUE,\n nudge_x = .3, nudge_y = .3\n ) +\n geom_point(data = bhat) +\n xlab(bquote(beta[1])) +\n ylab(bquote(beta[2])) +\n geom_label(\n data = bhat, mapping = aes(label = bquote(\"hat(beta)[s]^L\")), parse = TRUE,\n nudge_x = -.4, nudge_y = -.4\n )\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/unnamed-chunk-1-1.svg){fig-align='center'}\n:::\n:::\n\n\nThis regularization set...\n\n* ... is convex (computationally efficient)\n* ... has corners (performs variable selection)\n\n\n## $\\ell_1$-regularized regression\n\nKnown as \n\n* \"lasso\"\n* \"basis pursuit\"\n\nThe estimator satisfies\n\n$$\\blt = \\argmin_{ \\snorm{\\beta}_1 \\leq s} \\frac{1}{n}\\snorm{\\y-\\X\\beta}_2^2$$\n\n\nIn its corresponding Lagrangian dual form:\n\n$$\\bll = \\argmin_{\\beta} \\frac{1}{n}\\snorm{\\y-\\X\\beta}_2^2 + \\lambda \\snorm{\\beta}_1$$\n\n\n## Lasso\n\nWhile the ridge solution can be easily computed \n\n$$\\brl = \\argmin_{\\beta} \\frac 1n \\snorm{\\y-\\X\\beta}_2^2 + \\lambda \\snorm{\\beta}_2^2 = (\\X^{\\top}\\X + \\lambda \\mathbf{I})^{-1} \\X^{\\top}\\y$$\n\n\nthe lasso solution\n\n\n$$\\bll = \\argmin_{\\beta} \\frac 1n\\snorm{\\y-\\X\\beta}_2^2 + \\lambda \\snorm{\\beta}_1 = \\; ??$$\n\ndoesn't have a closed form solution.\n\n\nHowever, because the optimization problem is convex, there exist efficient algorithms for computing it\n\n(The best are Iterative Soft Thresholding or Coordinate Descent. Gradient Descent doesn't work very well in practice.)\n\n\n## Coefficient path: ridge vs lasso\n\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code code-fold=\"true\"}\nlibrary(glmnet)\ndata(prostate, package = \"ElemStatLearn\")\nX <- prostate %>% dplyr::select(-train,-lpsa) %>% as.matrix()\nY <- prostate$lpsa\nlasso <- glmnet(x = X, y = Y) # alpha = 1 by default\nridge <- glmnet(x = X, y = Y, alpha = 0)\nop <- par()\npar(mfrow = c(1, 2), mar = c(5, 3, 3, .1))\nplot(lasso, main = \"Lasso\")\nplot(ridge, main = \"Ridge\")\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/ridge-v-lasso-1.svg){fig-align='center'}\n:::\n:::\n\n::: {.cell layout-align=\"center\"}\n\n:::\n\n\n\n\n\n## Same but against Lambda\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\npar(mfrow = c(1, 2), mar = c(5, 3, 3, .1))\nplot(lasso, main = \"Lasso\", xvar = \"lambda\")\nplot(ridge, main = \"Ridge\", xvar = \"lambda\")\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/ridge-v-lasso-again-1.svg){fig-align='center'}\n:::\n:::\n\n::: {.cell layout-align=\"center\"}\n\n:::\n\n\n## Why does Lasso select variables?\n\nSuppose I have solutions for $\\widehat{\\beta}_j$, $j = 1,\\ldots,j-1, j+1,\\ldots,p$.\n\nLet $\\widehat{\\y}_{-j} = \\X_{-j}\\widehat{\\beta}_{-j}$.\n\nOne can show that:\n\n$$\n\\widehat{\\beta}_j = S\\left(\\mathbf{X}^\\top_j(\\y - \\widehat{\\y}_{-j}),\\ \\lambda\\right).\n$$\n\n$$\nS(z, \\gamma) = \\textrm{sign}(z)(|z| - \\gamma)_+ = \\begin{cases} z - \\gamma & z > \\gamma\\\\\nz + \\gamma & z < -\\gamma \\\\ 0 & |z| \\leq \\gamma \\end{cases}\n$$\n\nIterating over this is called [coordinate descent]{.secondary} and gives the solution.\n\n::: aside\nSee for example, \n:::\n\n\n::: notes\n* If I were told all the other coefficient estimates.\n* Then to find this one, I'd shrink when the gradient is big, or set to 0 if it\ngets too small.\n:::\n\n## Packages\n\nThere are two main `R` implementations for finding lasso\n\n\n`{glmnet}`: `lasso.out = glmnet(X, Y, alpha=1)`. \n\n* Setting `alpha = 0` gives ridge regression (as does `lm.ridge` in the `MASS` package)\n* Setting `alpha` $\\in (0,1)$ gives a method called the \"elastic net\" which combines ridge regression and lasso, more on that next lecture.\n* If you don't specify `alpha`, it does lasso\n\n`{lars}`: `lars.out = lars(X, Y)`\n* `lars` also does other things called \"Least angle\" and \"forward stagewise\" in addition to \"forward stepwise\" regression\n\n\n## (lots of others, but these are the biggies)\n\n1. `lars` (this one came first)\n\n2. `glmnet` (this one is faster)\n\nUse different algorithms, but both compute the solution for a range of $\\lambda$.\n\n`lars` starts with an empty model and adds coefficients until saturated. The sequence of $\\lambda$'s comes from the nature of the optimization problem.\n\n`glmnet` starts with an empty model and examines each value of $\\lambda$ using previous values as \"warm starts\". It is generally much faster than `lars` and uses lots of other tricks (as well as compiled code) for extra speed.\n\nThe path returned by `lars` is more useful than that returned by `glmnet`.\n\n. . .\n\nBut you should use `glmnet`.\n\n\n\n\n\n\n\n## Choosing the lambda\n\nYou have to choose $\\lambda$ in lasso or in ridge regression\n\nlasso selects variables (by setting coefficients to zero), but the value of $\\lambda$ determines how many/which.\n\nAll of these packages come with CV built in.\n\nHowever, the way to do it differs from package to package\n\n. . .\n\n

\n\n\n\n## `glmnet` version (lasso or ridge)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\n# 1. Estimate cv and model at once, no formula version\nlasso.glmnet <- cv.glmnet(X, Y) \n# no good reason to call glmnet() itself\n# 2. Look at the CV curve\n# 3. If the dashed lines are at the boundaries, redo with better lambda\nbest.lam <- lasso.glmnet$lambda.min \n# the value, not the location (or use lasso$lambda.1se)\n# 4. Return the coefs/predictions for the best model\ncoefs.glmnet <- coefficients(lasso.glmnet, s = \"lambda.min\")\npreds.glmnet <- predict(lasso.glmnet, newx = X, s = \"lambda.1se\") \n# must supply `newx`\n```\n:::\n\n\n* $\\widehat{R}_{CV}$ is an estimator of $R_n$, it has bias and variance\n* Because we did CV, we actually have 10 $\\widehat{R}$ values, 1 per split.\n* Calculate the mean (that's what we've been using), but what about SE?\n\n## `glmnet` version (lasso or ridge)\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\npar(mfrow = c(1, 2), mar = c(5, 3, 3, 0))\nplot(lasso.glmnet) # a plot method for the cv fit\nplot(lasso.glmnet$glmnet.fit) # the glmnet.fit == glmnet(X,Y)\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/unnamed-chunk-5-1.svg){fig-align='center'}\n:::\n:::\n\n\n\n\n## Paths with chosen lambda\n\n\n::: {.cell layout-align=\"center\"}\n\n```{.r .cell-code}\nridge.glmnet <- cv.glmnet(X, Y, alpha = 0, lambda.min.ratio = 1e-10) # added to get a minimum\npar(mfrow = c(1, 4))\nplot(ridge.glmnet, main = \"Ridge\")\nplot(lasso.glmnet, main = \"Lasso\")\nplot(ridge.glmnet$glmnet.fit, main = \"Ridge\")\nabline(v = sum(abs(coef(ridge.glmnet)))) # defaults to `lambda.1se`\nplot(lasso.glmnet$glmnet.fit, main = \"Lasso\")\nabline(v = sum(abs(coef(lasso.glmnet)))) # again, `lambda.1se` unless told otherwise\n```\n\n::: {.cell-output-display}\n![](09-l1-penalties_files/figure-revealjs/unnamed-chunk-7-1.svg){fig-align='center'}\n:::\n:::\n\n\n---\n\n## Degrees of freedom\n\nRemember: Lasso is **not** a linear smoother. There is no matrix $S$ such that $\\widehat{y} = Sy$ for the predicted values from lasso.\n\n* We can't use `cv_nice()`.\n\n* We don't have $\\tr{S} = \\textrm{df}$ because there is no $S$.\n\nHowever,\n\n* One can show that $\\textrm{df}_\\lambda = \\E[\\#(\\widehat{\\beta}_\\lambda \\neq 0)] = \\E[||\\widehat{\\beta}_\\lambda||_0]$\n\n* The proof is PhD-level material\n\nNote that the $\\widehat{\\textrm{df}}_\\lambda$ is plotted on the CV plot and that `lasso.glmnet$glmnet.fit$df` contains this value for all $\\lambda$.\n\n## Other flavours\n\nThe elastic net\n: generally used for correlated variables that\ncombines a ridge/lasso penalty. Included in `{glmnet}` (0 < `alpha` < 1). \n\nGrouped lasso\n: where variables are included or excluded in groups. Required for factors (1-hot encoding)\n\nRelaxed lasso\n: Takes the estimated model from lasso and fits the full least squares solution on the selected covariates (less bias, more variance). Included in `glmnet` with `relax = TRUE`\n\nDantzig selector\n: a slightly modified version of the lasso\n\n## Lasso cinematic universe\n\n::: flex\n::: w-60\n\nSCAD\n: a non-convex version of lasso that adds a more severe variable selection penalty\n\n$\\sqrt{\\textrm{lasso}}$\n: claims to be tuning parameter free (but isn't). Uses $||\\cdot||_2$\ninstead of $||\\cdot||_2^2$ for the loss.\n\nGeneralized lasso\n: Adds various additional matrices to the penalty term (ie: $||D\\beta||_1$. \n\n:::\n\n::: w-40\n\n![](https://sportshub.cbsistatic.com/i/2022/08/10/d348f903-585f-4aa6-aebc-d05173761065/brett-goldstein-hercules.jpg)\n\n:::\n:::\n\n\n# Next time...\n\nWhat happens when we're tired of all this linearity.\n", + "supporting": [ + "09-l1-penalties_files" + ], + "filters": [ + "rmarkdown/pagebreak.lua" + ], + "includes": { + "include-after-body": [ + "\n\n\n" + ] + }, + "engineDependencies": {}, + "preserve": {}, + "postProcess": true + } +} \ No newline at end of file diff --git a/_freeze/schedule/slides/09-l1-penalties/figure-revealjs/convexity-1.svg b/_freeze/schedule/slides/09-l1-penalties/figure-revealjs/convexity-1.svg new file mode 100644 index 0000000..7291a50 --- /dev/null +++ b/_freeze/schedule/slides/09-l1-penalties/figure-revealjs/convexity-1.svg @@ -0,0 +1,1185 @@ + + + + + + + + + + + + + + + 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+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/schedule/slides/08-ridge-regression.qmd b/schedule/slides/08-ridge-regression.qmd index 1ff6543..958fa6e 100644 --- a/schedule/slides/08-ridge-regression.qmd +++ b/schedule/slides/08-ridge-regression.qmd @@ -93,91 +93,6 @@ $$ \quad \st \beta \in \R^p $$ -## Visualizing ridge regression (2 coefficients) - -```{r} -#| code-fold: true -#| fig-width: 16 -#| fig-height: 9 -#| dev: png -b <- c(1, 1) -n <- 1000 -lams <- c(1, 5, 10) -ols_loss <- function(b1, b2) colMeans((y - X %*% rbind(b1, b2))^2) / 2 -pen <- function(b1, b2, lambda = 1) sqrt(n) * lambda * (b1^2 + b2^2) / 2 -gr <- expand_grid( - b1 = seq(b[1] - 0.5, b[1] + 0.5, length.out = 100), - b2 = seq(b[2] - 0.5, b[2] + 0.5, length.out = 100) -) - -X <- mvtnorm::rmvnorm(n, c(0, 0), sigma = matrix(c(1, .3, .3, .5), nrow = 2)) -y <- drop(X %*% b + rnorm(n)) - -bols <- coef(lm(y ~ X - 1)) -bridge <- coef(MASS::lm.ridge(y ~ X - 1, lambda = lams * sqrt(n))) - -penalties <- lams |> - set_names(~ paste("lam =", .)) |> - map(~ pen(gr$b1, gr$b2, .x)) |> - as_tibble() -gr <- gr |> - mutate(loss = ols_loss(b1, b2)) |> - bind_cols(penalties) - -g1 <- ggplot(gr, aes(b1, b2)) + - geom_raster(aes(fill = loss)) + - scale_fill_viridis_c(direction = -1) + - geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + - theme(legend.position = "bottom") + - guides(fill = guide_colourbar(barwidth = 20, barheight = 0.5)) - -g2 <- gr |> - pivot_longer(starts_with("lam")) |> - mutate(name = factor(name, levels = paste("lam =", lams))) |> - ggplot(aes(b1, b2)) + - geom_raster(aes(fill = value)) + - scale_fill_viridis_c(direction = -1, name = "penalty") + - facet_wrap(~name, ncol = 1) + - geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + - theme(legend.position = "bottom") + - guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5)) - -g3 <- gr |> - mutate(across(starts_with("lam"), ~ loss + .x)) |> - pivot_longer(starts_with("lam")) |> - mutate(name = factor(name, levels = paste("lam =", lams))) |> - ggplot(aes(b1, b2)) + - geom_raster(aes(fill = value)) + - scale_fill_viridis_c(direction = -1, name = "loss + pen") + - facet_wrap(~name, ncol = 1) + - geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + - theme(legend.position = "bottom") + - guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5)) - -cowplot::plot_grid(g1, g2, g3, rel_widths = c(2, 1, 1), nrow = 1) -``` - -## The effect on the estimates - -```{r} -#| code-fold: true -#| fig-width: 8 -#| fig-height: 5 -#| dev: png -gr |> - mutate(z = ols_loss(b1, b2) + max(lams) * pen(b1, b2)) |> - ggplot(aes(b1, b2)) + - geom_raster(aes(fill = z)) + - scale_fill_viridis_c(direction = -1) + - geom_point(data = tibble( - b1 = c(bols[1], bridge[,1]), - b2 = c(bols[2], bridge[,2]), - estimate = factor(c("ols", paste0("ridge = ", lams)), - levels = c("ols", paste0("ridge = ", lams))) - ), - aes(shape = estimate), size = 3) + - geom_point(data = data.frame(b1 = b[1], b2 = b[2]), colour = orange, size = 4) -``` ## Geometry of ridge regression (contours) @@ -268,6 +183,12 @@ $\ell_1$-norm (special case) $\ell_0$-norm : $\sum_{j=1}^p I(z_j \neq 0 ) = \lvert \{j : z_j \neq 0 \}\rvert$ +$\ell_\infty$-norm +: $\max_{1\leq j \leq p} |z_j|$ + +::: aside +Recall what a norm is: +::: ## Ridge regression @@ -304,6 +225,93 @@ Observe: $\lambda$ and $s$ are known as [tuning parameters]{.secondary} +## Visualizing ridge regression (2 coefficients) + +```{r} +#| code-fold: true +#| fig-width: 16 +#| fig-height: 9 +#| dev: png +b <- c(1, 1) +n <- 1000 +lams <- c(1, 5, 10) +ols_loss <- function(b1, b2) colMeans((y - X %*% rbind(b1, b2))^2) / 2 +pen <- function(b1, b2, lambda = 1) lambda * (b1^2 + b2^2) / 2 +gr <- expand_grid( + b1 = seq(b[1] - 0.5, b[1] + 0.5, length.out = 100), + b2 = seq(b[2] - 0.5, b[2] + 0.5, length.out = 100) +) + +X <- mvtnorm::rmvnorm(n, c(0, 0), sigma = matrix(c(1, .3, .3, .5), nrow = 2)) +y <- drop(X %*% b + rnorm(n)) + +bols <- coef(lm(y ~ X - 1)) +bridge <- coef(MASS::lm.ridge(y ~ X - 1, lambda = lams * sqrt(n))) + +penalties <- lams |> + set_names(~ paste("lam =", .)) |> + map(~ pen(gr$b1, gr$b2, .x)) |> + as_tibble() +gr <- gr |> + mutate(loss = ols_loss(b1, b2)) |> + bind_cols(penalties) + +g1 <- ggplot(gr, aes(b1, b2)) + + geom_raster(aes(fill = loss)) + + scale_fill_viridis_c(direction = -1) + + geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + + theme(legend.position = "bottom") + + guides(fill = guide_colourbar(barwidth = 20, barheight = 0.5)) + +g2 <- gr |> + pivot_longer(starts_with("lam")) |> + mutate(name = factor(name, levels = paste("lam =", lams))) |> + ggplot(aes(b1, b2)) + + geom_raster(aes(fill = value)) + + scale_fill_viridis_c(direction = -1, name = "penalty") + + facet_wrap(~name, ncol = 1) + + geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + + theme(legend.position = "bottom") + + guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5)) + +g3 <- gr |> + mutate(across(starts_with("lam"), ~ loss + .x)) |> + pivot_longer(starts_with("lam")) |> + mutate(name = factor(name, levels = paste("lam =", lams))) |> + ggplot(aes(b1, b2)) + + geom_raster(aes(fill = value)) + + scale_fill_viridis_c(direction = -1, name = "loss + pen") + + facet_wrap(~name, ncol = 1) + + geom_point(data = data.frame(b1 = b[1], b2 = b[2])) + + theme(legend.position = "bottom") + + guides(fill = guide_colourbar(barwidth = 10, barheight = 0.5)) + +cowplot::plot_grid(g1, g2, g3, rel_widths = c(2, 1, 1), nrow = 1) +``` + +## The effect on the estimates + +```{r} +#| code-fold: true +#| fig-width: 8 +#| fig-height: 5 +#| dev: png +gr |> + mutate(z = ols_loss(b1, b2) + max(lams) * pen(b1, b2)) |> + ggplot(aes(b1, b2)) + + geom_raster(aes(fill = z)) + + scale_fill_viridis_c(direction = -1) + + geom_point(data = tibble( + b1 = c(bols[1], bridge[,1]), + b2 = c(bols[2], bridge[,2]), + estimate = factor(c("ols", paste0("ridge = ", lams)), + levels = c("ols", paste0("ridge = ", lams))) + ), + aes(shape = estimate), size = 3) + + geom_point(data = data.frame(b1 = b[1], b2 = b[2]), colour = orange, size = 4) +``` + + ## Example data `prostate` data from [ESL] @@ -330,12 +338,14 @@ library(glmnet) ridge <- glmnet(x = X, y = Y, alpha = 0, lambda.min.ratio = .00001) ``` +
+ ::: flex ::: w-60 ```{r} #| fig-width: 8 #| fig-height: 6 -plot(ridge, xvar = "lambda") +plot(ridge, xvar = "lambda", lwd = 3) ``` ::: @@ -363,7 +373,8 @@ $$\brl = (\X^\top\X + \lambda \mathbf{I})^{-1}\X^\top \y$$ * This is easy to calculate in `R` for any $\lambda$. -* However, computations and interpretation are simplified if we examine the Singular Value Decomposition of $\X = \mathbf{UDV}^\top$. +* However, computations and interpretation are simplified if we examine the +[Singular Value Decomposition]{.secondary} of $\X = \mathbf{UDV}^\top$. * Recall: any matrix has an SVD. @@ -431,6 +442,20 @@ Conclusion Aside : Engineering approach to solving linear systems is to always do this with small $\lambda$. The thinking is about the numerics rather than the statistics. +### Which $\lambda$ to use? + +Computational +: Use CV and pick the $\lambda$ that makes this smallest. + +Intuition (bias) +: As $\lambda\rightarrow\infty$, bias ⬆ + +Intuition (variance) +: As $\lambda\rightarrow\infty$, variance ⬇ + +You should think about why. + + ## Can we get the best of both worlds? @@ -446,7 +471,9 @@ To recap: * Ridge regression does not perform variable selection. -* But [picking]{.hand} $\lambda=3.7$ and thereby .hand[deciding] to predict with $\widehat{\beta}^R_{3.7}$ is [model selection]{.secondary}. +* But [picking]{.hand} $\lambda=3.7$ and thereby [deciding]{.hand} to predict with $\widehat{\beta}^R_{3.7}$ is [model selection]{.secondary}. + + ## Can we get the best of both worlds? diff --git a/schedule/slides/09-l1-penalties.Rmd b/schedule/slides/09-l1-penalties.Rmd deleted file mode 100644 index 6a5e0b9..0000000 --- a/schedule/slides/09-l1-penalties.Rmd +++ /dev/null @@ -1,370 +0,0 @@ ---- -title: "09 L1 penalties" -author: - - "STAT 406" - - "Daniel J. McDonald" -date: 'Last modified - `r Sys.Date()`' ---- - -```{r setup, include=FALSE, warning=FALSE, message=FALSE} -source("rmd_config.R") -``` - - -```{r css-extras, file="css-extras.R", echo=FALSE} -``` - - -## Last time - -$$\newcommand{\Expect}[1]{\mathbb{E}\left[ #1 \right]} -\newcommand{\Var}[1]{\mathbb{V}\left[ #1 \right]} -\newcommand{\Cov}[2]{\mathrm{Cov}\left[#1,\ #2\right]} -\newcommand{\given}{\ \vert\ } -\newcommand{\E}{\mathbb{E}} -\renewcommand{\P}{\mathbb{P}} -\newcommand{\R}{\mathbb{R}} -\newcommand{\tr}[1]{\mbox{tr}(#1)} -\newcommand{\brt}{\widehat{\beta}^R_{s}} -\newcommand{\brl}{\widehat{\beta}^R_{\lambda}} -\newcommand{\bls}{\widehat{\beta}_{ols}} -\newcommand{\blt}{\widehat{\beta}^L_{s}} -\newcommand{\bll}{\widehat{\beta}^L_{\lambda}} -\newcommand{\X}{\mathbf{X}} -\newcommand{\y}{\mathbf{y}}$$ - - - -Ridge regression: $\min \frac{1}{n}||\y-\X\beta||_2^2 \textrm{ subject to } ||\beta||_2^2 \leq s$ - -Best (in sample) linear regression model of size $s$: $\min \frac 1n ||\y-\X\beta||_2^2 \textrm{ subject to } ||\beta||_0 \leq s$ - -$||\beta||_0$ is the number of nonzero elements in $\beta$ - -Finding the "best" linear model (of size $s$, among these predictors, in sample) is a nonconvex optimization problem (In fact, it is NP-hard) - -Ridge regression is convex (easy to solve), but doesn't do variable selection - -Can we somehow "interpolate" to get both? - - - ---- - -## Geometry of convexity - -```{r plotting-functions, echo=FALSE} -library(mvtnorm) -normBall <- function(q=1, len=1000){ - tg = seq(0,2*pi, length=len) - out = data.frame(x = cos(tg)) %>% - mutate(b=(1-abs(x)^q)^(1/q), bm=-b) %>% - gather(key='lab', value='y',-x) - out$lab = paste0('"||" * beta * "||"', '[',signif(q,2),']') - return(out) -} - -ellipseData <- function(n=100,xlim=c(-2,3),ylim=c(-2,3), - mean=c(1,1), Sigma=matrix(c(1,0,0,.5),2)){ - df = expand.grid(x=seq(xlim[1],xlim[2],length.out = n), - y=seq(ylim[1],ylim[2],length.out = n)) - df$z = dmvnorm(df, mean, Sigma) - df -} -lballmax <- function(ed,q=1,tol=1e-6){ - ed = filter(ed, x>0,y>0) - for(i in 1:20){ - ff = abs((ed$x^q+ed$y^q)^(1/q)-1)0) break - tol = 2*tol - } - best = ed[ff,] - best[which.max(best$z),] -} -``` - - -```{r convexity,echo=FALSE, dev="svg", fig.height=7,fig.width=13,fig.align="center"} -nbs = list() -nbs[[1]] = normBall(0,1) -qs = c(.5,.75,1,1.5,2) -for(ii in 2:6) nbs[[ii]] = normBall(qs[ii-1]) -nbs = bind_rows(nbs) -nbs$lab = factor(nbs$lab, levels = unique(nbs$lab)) -seg = data.frame(lab=levels(nbs$lab)[1], - x0=c(-1,0),x1=c(1,0),y0=c(0,-1),y1=c(0,1)) -levels(seg$lab) = levels(nbs$lab) -ggplot(nbs, aes(x,y)) + geom_path(size=1.2) + - facet_wrap(~lab,labeller = label_parsed) + - geom_segment(data=seg,aes(x=x0,xend=x1,y=y0,yend=y1),size=1.2) + - theme_bw(base_family = "", base_size = 24) + - coord_equal() + geom_vline(xintercept = 0,size=.5) + - geom_hline(yintercept = 0,size=.5) + - xlab(bquote(beta[1])) + ylab(bquote(beta[2])) -``` - ---- - -## The best of both worlds - - -```{r, echo=FALSE, warning=FALSE, dev="svg", fig.height=4,fig.width=4,fig.align="center"} -nb = normBall(1) -ed = ellipseData() -bols = data.frame(x=1,y=1) -bhat = lballmax(ed, 1) -ggplot(nb,aes(x,y)) + xlim(-2,2) + ylim(-2,2) + geom_path(color=red) + - geom_contour(mapping=aes(z=z), color=blue, data=ed, bins=7) + - geom_vline(xintercept = 0) + geom_hline(yintercept = 0) + - geom_point(data=bols) + coord_equal() + - theme_bw(base_family = "", base_size = 24) + - geom_label(data=bols, mapping=aes(label=bquote('hat(beta)[ols]')), parse=TRUE, - nudge_x = .3, nudge_y = .3) + - geom_point(data=bhat) + xlab(bquote(beta[1])) + ylab(bquote(beta[2])) + - geom_label(data=bhat, mapping=aes(label=bquote('hat(beta)[s]^L')), parse=TRUE, - nudge_x = -.4, nudge_y = -.4) -``` - -This regularization set... - -* ... is convex (computationally efficient) -* ... has corners (performs variable selection) - ---- - -## $\ell_1$-regularized regression - -Known as - -* "lasso" -* "basis pursuit" - -The estimator satisfies - -$$\blt = \arg\min_{ ||\beta||_1 \leq s} \frac{1}{n}||\y-\X\beta||_2^2$$ - - -In its corresponding Lagrangian dual form: - -$$\bll = \arg\min_{\beta} \frac{1}{n}||\y-\X\beta||_2^2 + \lambda ||\beta||_1$$ - ---- - -## Lasso - -While the ridge solution can be easily computed - -$$\brl = \arg\min_{\beta} \frac 1n ||\y-\X\beta||_2^2 + \lambda ||\beta||_2^2 = (\X^{\top}\X + \lambda \mathbf{I})^{-1} \X^{\top}\y$$ - - -the lasso solution - - -$$\bll = \arg\min_{\beta} \frac 1n||\y-\X\beta||_2^2 + \lambda ||\beta||_1 = \; ??$$ - -doesn't have a closed form solution. - - -However, because the optimization problem is convex, there exist efficient algorithms for computing it - -(The best are Iterative Soft Thresholding or Coordinate Descent. Gradient Descent doesn't work very well in practice.) - ---- - -## Coefficient path: ridge vs lasso - - -```{r ridge-v-lasso,echo=TRUE,dev="svg",fig.align="center", fig.width=11, fig.height=4} -library(glmnet) -data(prostate, package = "ElemStatLearn") -X <- prostate %>% dplyr::select(-train,-lpsa) %>% as.matrix() -Y <- prostate$lpsa -lasso <- glmnet(x = X, y = Y) # alpha = 1 by default -ridge <- glmnet(x = X, y = Y, alpha = 0) -op <- par() -par(mfrow = c(1, 2), mar = c(5,3,3,.1)) -plot(lasso, main = "Lasso") -plot(ridge, main = "Ridge") -par(op) -``` - -```{r tidy-glmnet, include = FALSE, eval = FALSE} -df = data.frame(as.matrix(t(ridge$beta))) -df1 = data.frame(as.matrix(t(lasso$beta))) -df$l1norm = colSums(abs(ridge$beta)) -df1$l1norm = colSums(abs(lasso$beta)) -df$method = 'ridge' -df1$method = 'lasso' -bind_rows(df,df1) %>% - pivot_longer(names_to='predictor',values_to ='coefficient', - cols=-c(l1norm,method)) %>% - ggplot(aes(x=l1norm, y=coefficient, color=predictor)) + geom_path() + - facet_wrap(~method,scales = 'free_x') + - geom_hline(color="black",linetype="dotted",yintercept = 0) + - scale_color_brewer(palette = 'Set1') + theme_cowplot() -``` - ---- - -## Same but against Lambda - -```{r ridge-v-lasso-again, fig.width=11, fig.height=5} -op <- par() -par(mfrow = c(1, 2), mar = c(5,3,3,.1)) -plot(lasso, main = "Lasso", xvar = "lambda") -plot(ridge, main = "Ridge", xvar = "lambda") -par(op) - -``` - ---- - -## Packages - -There are two main `R` implementations for finding lasso - - -`{glmnet}`: `lasso.out = glmnet(X, Y, alpha=1)`. - -* Setting `alpha = 0` gives ridge regression (as does `lm.ridge` in the `MASS` package) -* Setting `alpha` $\in (0,1)$ gives a method called the "elastic net" which combines ridge regression and lasso, more on that next lecture. -* If you don't specify `alpha`, it does lasso - -`{lars}`: `lars.out = lars(X, Y)` -* `lars` also does other things called "Least angle" and "forward stagewise" in addition to "forward stepwise" regression - ---- - -## (lots of others, but these are the biggies) - -1. `lars` (this one came first) - -2. `glmnet` (this one is faster) - -Use different algorithms, but both compute the solution for a range of $\lambda$. - -`lars` starts with an empty model and adds coefficients until saturated. The sequence of $\lambda$'s comes from the nature of the optimization problem. - -`glmnet` starts with an empty model and examines each value of $\lambda$ using previous values as "warm starts". It is generally much faster than `lars` and uses lots of other tricks (as well as compiled code) for extra speed. - -The path returned by `lars` is more useful than that returned by `glmnet`. - --- - -But you should use `glmnet`. - - - - ---- - - - -## Choosing the lambda - -You have to choose $\lambda$ in lasso or in ridge regression - -lasso selects variables (by setting coefficients to zero), but the value of $\lambda$ determines how many/which. - -All of these packages come with CV built in. - -However, the way to do it differs from package to package - --- - -

- ---- - - -## `glmnet` version (lasso or ridge) - -.pull-left[ -```{r} -# 1. Estimate cv and model at once -# no formula version -lasso.glmnet <- cv.glmnet(X, Y) -# no good reason to call glmnet() itself -# 2. Look at the CV curve -# 3. If the dashed lines are at the -# boundaries, redo with better lambda -best.lam <- lasso.glmnet$lambda.min -# the value, not the location -# (or use lasso$lambda.1se) -# 4. Return the coefs/predictions -# for the best model -coefs.glmnet <- coefficients( - lasso.glmnet, s = "lambda.min" -) - -preds.glmnet <- predict( - lasso.glmnet, newx = X, s = "lambda.1se" -) -# must supply `newx` -``` - -* $\widehat{R}_{CV}$ is an estimator of $R_n$, it has bias and variance -* Because we did CV, we actually have 10 $\widehat{R}$ values, 1 per split. -* Calculate the mean (that's what we've been using), but what about SE? -] - -.pull-right[ -```{r, dev="svg", fig.align="center",fig.width=4, fig.height=5} -par(mfrow=c(2,1), mar=c(5, 3, 3, 0)) -plot(lasso.glmnet) # a plot method for the cv fit -plot(lasso.glmnet$glmnet.fit) # the glmnet.fit == glmnet(X,Y) -``` - -```{r, include=FALSE} -par(op) -``` -] - ---- - -## Paths with chosen lambda - -```{r, fig.width=11,fig.align="center",dev="svg",fig.height=4} -ridge.glmnet <- cv.glmnet(X, Y, alpha = 0, lambda.min.ratio = 1e-10) # added to get a minimum -par(mfrow = c(1, 4)) -plot(ridge.glmnet, main = "Ridge") -plot(lasso.glmnet, main = "Lasso") -plot(ridge.glmnet$glmnet.fit, main = "Ridge") -abline(v = sum(abs(coef(ridge.glmnet)))) # defaults to `lambda.1se` -plot(lasso.glmnet$glmnet.fit, main = "Lasso") -abline(v = sum(abs(coef(lasso.glmnet)))) # again, `lambda.1se` unless told otherwise -``` - ---- - -## Degrees of freedom - -Remember: Lasso is **not** a linear smoother. There is no matrix $S$ such that $\widehat{y} = Sy$ for the predicted values from lasso. - -* We can't use `cv_nice()`. - -* We don't have $tr(S) = \textrm{df}$ because there is no $S$. - -However, - -* One can show that $\textrm{df}_\lambda = \#(\widehat{\beta}_\lambda \neq 0) = ||\widehat{\beta}_\lambda||_0$ - -* The proof is PhD-level material - -Note that the $\textrm{df}_\lambda$ is plotted on the CV plot and that `lasso.glmnet$glmnet.fit$df` contains this value for all $\lambda$. - --- - -One might suspect, then, that for Elastic Net - -$$\min_\beta \frac{1}{n} || \mathbf{y} - \mathbf{X}\beta||_2 + \alpha\lambda||\beta||_1 + (1-\alpha)\lambda||\beta||^2_2$$ - -We have $\textrm{df}_\lambda = \alpha ||\beta_\lambda||_0 + (1-\alpha) tr (S_\lambda)$ where $S$ comes from the ridge penalty. - -This is an OK approximation in practice, but not quite right. ---- -class: middle, inverse, center - -# Next time... - -What happens when we're tired of all this linearity. \ No newline at end of file diff --git a/schedule/slides/09-l1-penalties.html b/schedule/slides/09-l1-penalties.html deleted file mode 100644 index 29bb7e0..0000000 --- a/schedule/slides/09-l1-penalties.html +++ /dev/null @@ -1,482 +0,0 @@ - - - - 09 L1 penalties - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - diff --git a/schedule/slides/09-l1-penalties.qmd b/schedule/slides/09-l1-penalties.qmd new file mode 100644 index 0000000..0665b4d --- /dev/null +++ b/schedule/slides/09-l1-penalties.qmd @@ -0,0 +1,414 @@ +--- +lecture: "09 L1 penalties" +format: revealjs +metadata-files: + - _metadata.yml +--- + +{{< include _titleslide.qmd >}} + + +## Last time + + +Ridge regression +: $\min \frac{1}{n}\snorm{\y-\X\beta}_2^2 \st \snorm{\beta}_2^2 \leq s$ + +Best (in sample) linear regression model of size $s$ +: $\min \frac 1n \snorm{\y-\X\beta}_2^2 \st \snorm{\beta}_0 \leq s$ + +$\snorm{\beta}_0$ is the number of nonzero elements in $\beta$ + +Finding the "best" linear model (of size $s$, among these predictors, in sample) is a nonconvex optimization problem (In fact, it is NP-hard) + +Ridge regression is convex (easy to solve), but doesn't do variable selection + +Can we somehow "interpolate" to get both? + + + +## Geometry of convexity + +```{r plotting-functions} +#| code-fold: true +#| fig-width: 12 +#| fig-height: 6 +library(mvtnorm) +normBall <- function(q = 1, len = 1000) { + tg <- seq(0, 2 * pi, length = len) + out <- data.frame(x = cos(tg)) %>% + mutate(b = (1 - abs(x)^q)^(1 / q), bm = -b) %>% + gather(key = "lab", value = "y", -x) + out$lab <- paste0('"||" * beta * "||"', "[", signif(q, 2), "]") + return(out) +} + +ellipseData <- function(n = 100, xlim = c(-2, 3), ylim = c(-2, 3), + mean = c(1, 1), Sigma = matrix(c(1, 0, 0, .5), 2)) { + df <- expand.grid( + x = seq(xlim[1], xlim[2], length.out = n), + y = seq(ylim[1], ylim[2], length.out = n) + ) + df$z <- dmvnorm(df, mean, Sigma) + df +} + +lballmax <- function(ed, q = 1, tol = 1e-6) { + ed <- filter(ed, x > 0, y > 0) + for (i in 1:20) { + ff <- abs((ed$x^q + ed$y^q)^(1 / q) - 1) < tol + if (sum(ff) > 0) break + tol <- 2 * tol + } + best <- ed[ff, ] + best[which.max(best$z), ] +} + +nbs <- list() +nbs[[1]] <- normBall(0, 1) +qs <- c(.5, .75, 1, 1.5, 2) +for (ii in 2:6) nbs[[ii]] <- normBall(qs[ii - 1]) +nbs <- bind_rows(nbs) +nbs$lab <- factor(nbs$lab, levels = unique(nbs$lab)) +seg <- data.frame( + lab = levels(nbs$lab)[1], + x0 = c(-1, 0), x1 = c(1, 0), y0 = c(0, -1), y1 = c(0, 1) +) +levels(seg$lab) <- levels(nbs$lab) +ggplot(nbs, aes(x, y)) + + geom_path(size = 1.2) + + facet_wrap(~lab, labeller = label_parsed) + + geom_segment(data = seg, aes(x = x0, xend = x1, y = y0, yend = y1), size = 1.2) + + theme_bw(base_family = "", base_size = 24) + + coord_equal() + + scale_x_continuous(breaks = c(-1, 0, 1)) + + scale_y_continuous(breaks = c(-1, 0, 1)) + + geom_vline(xintercept = 0, size = .5) + + geom_hline(yintercept = 0, size = .5) + + xlab(bquote(beta[1])) + + ylab(bquote(beta[2])) +``` + + +## The best of both worlds + + +```{r} +#| code-fold: true +nb <- normBall(1) +ed <- ellipseData() +bols <- data.frame(x = 1, y = 1) +bhat <- lballmax(ed, 1) +ggplot(nb, aes(x, y)) + + geom_path(color = red) + + geom_contour(mapping = aes(z = z), color = blue, data = ed, bins = 7) + + geom_vline(xintercept = 0) + + geom_hline(yintercept = 0) + + geom_point(data = bols) + + coord_equal(xlim = c(-2, 2), ylim = c(-2, 2)) + + theme_bw(base_family = "", base_size = 24) + + geom_label( + data = bols, mapping = aes(label = bquote("hat(beta)[ols]")), parse = TRUE, + nudge_x = .3, nudge_y = .3 + ) + + geom_point(data = bhat) + + xlab(bquote(beta[1])) + + ylab(bquote(beta[2])) + + geom_label( + data = bhat, mapping = aes(label = bquote("hat(beta)[s]^L")), parse = TRUE, + nudge_x = -.4, nudge_y = -.4 + ) +``` + +This regularization set... + +* ... is convex (computationally efficient) +* ... has corners (performs variable selection) + + +## $\ell_1$-regularized regression + +Known as + +* "lasso" +* "basis pursuit" + +The estimator satisfies + +$$\blt = \argmin_{ \snorm{\beta}_1 \leq s} \frac{1}{n}\snorm{\y-\X\beta}_2^2$$ + + +In its corresponding Lagrangian dual form: + +$$\bll = \argmin_{\beta} \frac{1}{n}\snorm{\y-\X\beta}_2^2 + \lambda \snorm{\beta}_1$$ + + +## Lasso + +While the ridge solution can be easily computed + +$$\brl = \argmin_{\beta} \frac 1n \snorm{\y-\X\beta}_2^2 + \lambda \snorm{\beta}_2^2 = (\X^{\top}\X + \lambda \mathbf{I})^{-1} \X^{\top}\y$$ + + +the lasso solution + + +$$\bll = \argmin_{\beta} \frac 1n\snorm{\y-\X\beta}_2^2 + \lambda \snorm{\beta}_1 = \; ??$$ + +doesn't have a closed form solution. + + +However, because the optimization problem is convex, there exist efficient algorithms for computing it + +(The best are Iterative Soft Thresholding or Coordinate Descent. Gradient Descent doesn't work very well in practice.) + + +## Coefficient path: ridge vs lasso + + +```{r ridge-v-lasso} +#| code-fold: true +library(glmnet) +data(prostate, package = "ElemStatLearn") +X <- prostate %>% dplyr::select(-train,-lpsa) %>% as.matrix() +Y <- prostate$lpsa +lasso <- glmnet(x = X, y = Y) # alpha = 1 by default +ridge <- glmnet(x = X, y = Y, alpha = 0) +op <- par() +par(mfrow = c(1, 2), mar = c(5, 3, 3, .1)) +plot(lasso, main = "Lasso") +plot(ridge, main = "Ridge") +``` + +```{r} +#| echo: false +par(op) +``` + + +```{r tidy-glmnet} +#| include: false +#| eval: false +df <- data.frame(as.matrix(t(ridge$beta))) +df1 <- data.frame(as.matrix(t(lasso$beta))) +df$l1norm <- colSums(abs(ridge$beta)) +df1$l1norm <- colSums(abs(lasso$beta)) +df$method <- "ridge" +df1$method <- "lasso" +bind_rows(df, df1) %>% + pivot_longer( + names_to = "predictor", values_to = "coefficient", + cols = -c(l1norm, method) + ) %>% + ggplot(aes(x = l1norm, y = coefficient, color = predictor)) + + geom_path() + + facet_wrap(~method, scales = "free_x") + + geom_hline(color = "black", linetype = "dotted", yintercept = 0) + + scale_color_brewer(palette = "Set1") +``` + + +## Same but against Lambda + +```{r ridge-v-lasso-again} +par(mfrow = c(1, 2), mar = c(5, 3, 3, .1)) +plot(lasso, main = "Lasso", xvar = "lambda") +plot(ridge, main = "Ridge", xvar = "lambda") +``` + +```{r} +#| echo: false +par(op) +``` + +## Why does Lasso select variables? + +Suppose I have solutions for $\widehat{\beta}_j$, $j = 1,\ldots,j-1, j+1,\ldots,p$. + +Let $\widehat{\y}_{-j} = \X_{-j}\widehat{\beta}_{-j}$. + +One can show that: + +$$ +\widehat{\beta}_j = S\left(\mathbf{X}^\top_j(\y - \widehat{\y}_{-j}),\ \lambda\right). +$$ + +$$ +S(z, \gamma) = \textrm{sign}(z)(|z| - \gamma)_+ = \begin{cases} z - \gamma & z > \gamma\\ +z + \gamma & z < -\gamma \\ 0 & |z| \leq \gamma \end{cases} +$$ + +Iterating over this is called [coordinate descent]{.secondary} and gives the solution. + +::: aside +See for example, +::: + + +::: notes +* If I were told all the other coefficient estimates. +* Then to find this one, I'd shrink when the gradient is big, or set to 0 if it +gets too small. +::: + +## Packages + +There are two main `R` implementations for finding lasso + + +`{glmnet}`: `lasso.out = glmnet(X, Y, alpha=1)`. + +* Setting `alpha = 0` gives ridge regression (as does `lm.ridge` in the `MASS` package) +* Setting `alpha` $\in (0,1)$ gives a method called the "elastic net" which combines ridge regression and lasso, more on that next lecture. +* If you don't specify `alpha`, it does lasso + +`{lars}`: `lars.out = lars(X, Y)` +* `lars` also does other things called "Least angle" and "forward stagewise" in addition to "forward stepwise" regression + + +## (lots of others, but these are the biggies) + +1. `lars` (this one came first) + +2. `glmnet` (this one is faster) + +Use different algorithms, but both compute the solution for a range of $\lambda$. + +`lars` starts with an empty model and adds coefficients until saturated. The sequence of $\lambda$'s comes from the nature of the optimization problem. + +`glmnet` starts with an empty model and examines each value of $\lambda$ using previous values as "warm starts". It is generally much faster than `lars` and uses lots of other tricks (as well as compiled code) for extra speed. + +The path returned by `lars` is more useful than that returned by `glmnet`. + +. . . + +But you should use `glmnet`. + + + + + + + +## Choosing the lambda + +You have to choose $\lambda$ in lasso or in ridge regression + +lasso selects variables (by setting coefficients to zero), but the value of $\lambda$ determines how many/which. + +All of these packages come with CV built in. + +However, the way to do it differs from package to package + +. . . + +

+ + + +## `glmnet` version (lasso or ridge) + +```{r} +# 1. Estimate cv and model at once, no formula version +lasso.glmnet <- cv.glmnet(X, Y) +# no good reason to call glmnet() itself +# 2. Look at the CV curve +# 3. If the dashed lines are at the boundaries, redo with better lambda +best.lam <- lasso.glmnet$lambda.min +# the value, not the location (or use lasso$lambda.1se) +# 4. Return the coefs/predictions for the best model +coefs.glmnet <- coefficients(lasso.glmnet, s = "lambda.min") +preds.glmnet <- predict(lasso.glmnet, newx = X, s = "lambda.1se") +# must supply `newx` +``` + +* $\widehat{R}_{CV}$ is an estimator of $R_n$, it has bias and variance +* Because we did CV, we actually have 10 $\widehat{R}$ values, 1 per split. +* Calculate the mean (that's what we've been using), but what about SE? + +## `glmnet` version (lasso or ridge) + +```{r} +par(mfrow = c(1, 2), mar = c(5, 3, 3, 0)) +plot(lasso.glmnet) # a plot method for the cv fit +plot(lasso.glmnet$glmnet.fit) # the glmnet.fit == glmnet(X,Y) +``` + +```{r, include=FALSE} +par(op) +``` + +## Paths with chosen lambda + +```{r, fig.width=11,fig.align="center",dev="svg",fig.height=4} +ridge.glmnet <- cv.glmnet(X, Y, alpha = 0, lambda.min.ratio = 1e-10) # added to get a minimum +par(mfrow = c(1, 4)) +plot(ridge.glmnet, main = "Ridge") +plot(lasso.glmnet, main = "Lasso") +plot(ridge.glmnet$glmnet.fit, main = "Ridge") +abline(v = sum(abs(coef(ridge.glmnet)))) # defaults to `lambda.1se` +plot(lasso.glmnet$glmnet.fit, main = "Lasso") +abline(v = sum(abs(coef(lasso.glmnet)))) # again, `lambda.1se` unless told otherwise +``` + +--- + +## Degrees of freedom + +Remember: Lasso is **not** a linear smoother. There is no matrix $S$ such that $\widehat{y} = Sy$ for the predicted values from lasso. + +* We can't use `cv_nice()`. + +* We don't have $\tr{S} = \textrm{df}$ because there is no $S$. + +However, + +* One can show that $\textrm{df}_\lambda = \E[\#(\widehat{\beta}_\lambda \neq 0)] = \E[||\widehat{\beta}_\lambda||_0]$ + +* The proof is PhD-level material + +Note that the $\widehat{\textrm{df}}_\lambda$ is plotted on the CV plot and that `lasso.glmnet$glmnet.fit$df` contains this value for all $\lambda$. + +## Other flavours + +The elastic net +: generally used for correlated variables that +combines a ridge/lasso penalty. Included in `{glmnet}` (0 < `alpha` < 1). + +Grouped lasso +: where variables are included or excluded in groups. Required for factors (1-hot encoding) + +Relaxed lasso +: Takes the estimated model from lasso and fits the full least squares solution on the selected covariates (less bias, more variance). Included in `glmnet` with `relax = TRUE` + +Dantzig selector +: a slightly modified version of the lasso + +## Lasso cinematic universe + +::: flex +::: w-60 + +SCAD +: a non-convex version of lasso that adds a more severe variable selection penalty + +$\sqrt{\textrm{lasso}}$ +: claims to be tuning parameter free (but isn't). Uses $||\cdot||_2$ +instead of $||\cdot||_2^2$ for the loss. + +Generalized lasso +: Adds various additional matrices to the penalty term (ie: $||D\beta||_1$. + +::: + +::: w-40 + +![](https://sportshub.cbsistatic.com/i/2022/08/10/d348f903-585f-4aa6-aebc-d05173761065/brett-goldstein-hercules.jpg) + +::: +::: + + +# Next time... + +What happens when we're tired of all this linearity. diff --git a/schedule/slides/_titleslide.qmd b/schedule/slides/_titleslide.qmd index 3a6bea6..9b303f7 100644 --- a/schedule/slides/_titleslide.qmd +++ b/schedule/slides/_titleslide.qmd @@ -33,5 +33,11 @@ $$ \newcommand{\R}{\mathbb{R}} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \newcommand{\snorm}[1]{\lVert #1 \rVert} +\newcommand{\tr}[1]{\mbox{tr}(#1)} +\newcommand{\brt}{\widehat{\beta}^R_{s}} +\newcommand{\brl}{\widehat{\beta}^R_{\lambda}} +\newcommand{\bls}{\widehat{\beta}_{ols}} +\newcommand{\blt}{\widehat{\beta}^L_{s}} +\newcommand{\bll}{\widehat{\beta}^L_{\lambda}} $$

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